A SAMPLE PAPER

DANIEL R. GRAYSON

Introduction

Every math paper has an introduction that tells what the paper is about. It doesn’t contain any proofs.

1. Theorems and such things

In your expositions, please be aware of the meanings of words. For example, an equation is a pair of expressions separated by an equal sign, and one may ask whether an equation is true or false, or for which values of the variables it holds. An example of an equation is ax^{2} + bx + c = 0. A symbol is usually a single letter. Examples of symbols are x and c. If you use symbols as abbreviated notation for something else, then it’s a good idea to define them before using them. An expression is something (without an equal sign) which represents a mathematical entity, often a number. An example of an expression is b^{2} −4ac or x^{n+1}. A formula is an equation that provides a solution to another equation. An example of a formula is

x = \frac{−b ± \sqrt{b^{2} − 4ac}}{2a},

which provides a solution to the quadratic equation above. One proves a statement and one constructs a geometric figure. A term is a member of a sum, and a factor is a member of a product. Be careful not to start a sentence with a mathematical symbol, variable, or number.

A theorem is an important true statement that the reader may want to use or refer to. Theorem 1.1 below is an example.

A lemma is a mathematical fact worth presenting separately that plays a role in the proof of a later theorem. It’s usually not deemed to be of independent interest to the reader.

A proposition is intermediate between a lemma and a theorem. Perhaps it is useful in proving a theorem, and also of some independent interest.

Theorems, lemmas, and propositions, should come with proofs.

Theorem 1.1. If x = 1, then x + x = 2.

Proof. We compute as follows: x + x = 1 + 1 = 2. □

Date: August 22, 2003.

2. Citations

Some papers might even include references to the literature, such as [2] or [1, Theorem 1]. The bibliographic information, in bibtex format, can be downloaded from MathSciNet at http://e-math.ams.org/mathscinet/ and placed into a separate file we’ll call sample.bib for processing by bibtex.

You probably won’t need this in most of the papers for our class.

References

1. Wolfgang Krull, Jacobsonsches Radikal und Hilbertscher Nullstellensatz, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2 (Providence, R. I.), Amer. Math. Soc., 1952, pp. 56–64. MR 13,526c

2. Oscar Zariski, A new proof of Hilbert’s Nullstellensatz, Bull. Amer. Math. Soc. 53 (1947), 362–368. MR 8,499g

University of Illinois at Urbana-Champaign

E-mail address: dan@math.uiuc.edu
URL: http://www.math.uiuc.edu/~dan/